The subject of this article is a story which is both comic and sad. It is a story of how two people, of which one (Eliahu Rips) was a
brilliant mathematician with a record of excellent achievements in modern
mathematics and the other (Doron Witztum) although
without any scientific credentials, was obviously very smart and ingenious,
started what looked like a serious scientific research, and very soon, motivated
by obviously non-scientific goals, went astray so their activity degenerated to
propagating absurdity. The history
of science knows more than one example of alleged discoveries which turned out
to be the results of experimental errors. The
authors of these pseudo-discoveries reacted to the disproval of their claims in
various ways. Many of them promptly admitted their errors. Of course, this is
the most respectable way to deal with the situation and the best way to preserve
a good reputation. The heroes of
this story, however, continue to fight tooth and nail trying to assert the
validity of their theory despite all the proofs against it. This is also a story
of how an allegedly-scientific discovery, despite being shown to be
unsubstantiated, was appropriated with enthusiasm by a crowd of people who did
not understand the essence of the work by the theory's originators but were
delighted by its religious implications and vulgarized it to such an extent that
even its original authors had to assert the absurdity of the claims of their
epigones.

Statistical Science is a prestigious peer-reviewed scientific journal
published by The Institute of Mathematical Statistics. It prints papers on a
variety of subjects, dealing with the application of statistical methods to any
area of research. Its readership is
limited mainly to faculty and researchers interested in the use of mathematical
statistics in various fields of science and technology. Unexpectedly, though, one article [1] printed in that journal in 1994
found a much broader audience since it was reprinted in full in a best-selling
sensational book [2]. Its three
authors were Doron Witztum, Eliahu Rips and Yoav Rosenberg (to be referred as
WRR.)

The essence of WRR's paper was as follows. First, these writers showed that in the Hebrew text of Genesis
there are large numbers of what they called Equidistant Letter Sequences
(ELS.) This term denotes words
formed in a text by letters separated by equal intervals ("skips"). For example, look at the word "DenOteS"
in the preceding sentence. Letters
"D," "O,"
and "S" are separated by equal intervals of 2 letters (en
between D and O, and te
between O and S). The three equidistant letters form the acronym DOS,
which stands for Disk Operating System
and constitutes an ELS with a skip of
3. According to WRR, the ELS that run along the text (which in Hebrew is from
right to left) and against text's
flow (in Hebrew from left to right) are equally valid. In the latter case the skip is negative. For example, in the same word "denotes," if read from right to left,
letters s, o and d constitute an ELS spelling SOD with a
skip of -3. Naturally, every word
of the text itself is also an ELS with a skip of 1.

Furthermore, WRR claimed that in the text of Genesis there are multiple pairs of ELS, with the words of the pair related by meaning and
situated within the text in close
proximity to each other. WRR claimed that the proximity in question far
exceeds what could be expected if those pairs of ELS appeared in the text by
chance alone.

To prove their point, WRR conducted a computerized statistical experiment. They first compiled a list of 34 "famous rabbis" who
lived between early medieval times and the 18th century and whose
biographies in the Encyclopedia of Great
Men of Israel [3] each occupied at least three columns. Later, following
advice from a referee, they compiled a second list of 32 "less famous
rabbis," whose biographies each occupied between 1.5 and 3 columns. Then, with
the help of an expert in Judaic bibliography, Professor S. Havlin of the Bar-Ilan
University in Israel, they matched the rabbis' various appellations, with the
dates of those rabbis' birth and/or death, according to the Responsa
database maintained by the Bar-Ilan University. In Hebrew the dates are written using letters of the alphabet, so WRR
searched in the text of Genesis for ELS both for the appellations and for the
dates.

They suggested a formula which supposedly estimated the "distance" between
any two ELS and used it to "measure" distances between ELS for appellations
and ELS for dates. Then they shuffled the lists of appellations and of dates,
creating 999999 scrambled lists, where the rabbis' names and the pertinent
dates were randomly mismatched. Their
computer program measured the "distance" for each pair of
appellations/dates, both in the correct
and in the mismatched lists. Then the program calculated four aggregate measures
of "proximity" between appellations and dates for each one of the above one
million of lists. They denoted the aggregate criteria of proximity
as statistics P1 , P2 ,
P3, and P4. The
values of these proximity measures were then ranked, assigning rank 1 to the appellations/dates list which
displayed the "best" proximity (that is the lowest values of the above
statistics), rank 2 to the list with the next higher value of P etc. Somewhere on the ladder of ranks thus created was the original
"correct" list of appellations/dates.

The results claimed by WRR were nothing less than astonishing. WRR claimed that the ranks of the "correct" (unscrambled) list of
appellations/dates were always very low, never exceeding a few thousand out of
one million competing scrambled lists, and often much less than that. WRR concluded that the ELS for the rabbis' appellations and ELS for their correct dates of birth/death, are situated in the
text of Genesis at an unusually close
proximity, with the level of confidence 1 in 62,000. WRR finally stated that
the proximity of the ELS for the
rabbis' appellations to the ELS for their correct dates of birth/death in the
Book of Genesis was "not due to chance."

The meaning of such conclusion was quite far reaching. Since the Book of Genesis
was written many centuries before the rabbis in question were born, its creator
must have known the future. Then
the results by WRR, in their opinion (shared also by many others who were
impressed by the amazing outcome of WRR's experiment) prove the divine origin
of the Book of Genesis. Hence the
results of WRR's experiment were touted as the alleged scientific
proof of the existence of God, and more specifically, of the God of the Torah.

The editor of Statistical Science, R.
Kass supplied WRR's paper with a comment saying that the paper was being
published as a puzzle offered to readers. In
subsequent additional comments posted on the Internet Kass made it clear that
the editors did not accept WRR's claims, and hoped that some readers would be
willing to invest enough time and effort to unearth the hidden flaws in WRR's
methodology.

The publication of WRR's paper had a number of unexpected consequences. One consequence was the appearance of a large number of papers, lectures,
Internet postings, etc, whose authors, most of them not capable of comprehending
the mathematical apparatus in WRR's article, vulgarized WRR's results by
searching for multiple ELS in the Bible without any regard to a statistical
verification of their claims. A
number of computer programs enabling anyone to search for ELS in the Bible have
been peddled all over the Internet. Some
other fast entrepreneurs offer to find, for a fee, the information allegedly "encoded" in the Bible about anyone's personal matters. Many other writers and authors of Web postings, referring to WRR's
paper as "the proof" of the Bible code, utilize ELS in the Bible to promote
various agendas. One that seems to
be most widely spread is that of Christian preachers.

There is an organization named Aish Ha
Torah, whose aim is to attract those Jews who have either lost faith or have
never been observant back to the fold. This
is done by conducting seminars where trained lecturers discuss various facets of
Judaism. Even though various topics are presented in these lectures, the subject
of the "codes" in the Torah is strongly emphasized there, as one of the main
tools for convincing the doubting
participants to re-embrace the faith. That organization maintains a website (Torah Codes) in which its creators state that their goal is to provide a forum
to Witztum and to promote his work.

At least nine books about the "Bible code" have recently appeared, at
least two pseudo- "documentaries" have been broadcast, and at least one
dramatic film has been released, all of them promoting and supporting the Bible
code. The largest splash was
probably made by the sensational book [2] published by the journalist Michael
Drosnin and titled "The Bible Code." The book appeared on the New York Times list of best-sellers and was translated into many
languages, while its author was featured in Newsweek
and Time, appeared on a number of TV
shows, sold movie rights for the book to Warner Bros., and set out on a world
wide promotion tour, in the course of which he did not spare some not quite
polite words regarding those people who dared to doubt the validity of
Drosnin's sensational claims.

Drosnin claimed to have discovered in the Bible multiple predictions of future
events "encoded" as clusters of ELS. The
most loudly advertised was the alleged prediction of the assassination of
Israeli Prime Minister Itzhak Rabin.

In
Fig.1 a table from Drosnin's book is shown with the letters of an ELS that
spells the name Itzhak Rabin circled. The
skip of this ELS is 4772. In order to display this ELS on a vertical, the text of the Bible was
converted into a table 4772 letters wide (the picture shows a "window" into
that table, which contains the ELS in question as well as a set of letters of
the plain text, surrounded by rectangles, spelling words Rotzeakh
Asher Irtzakh, meaning "killer who will kill"). The table in fig.1 also
contains a number of ELS for words seemingly related by meaning to Rabin's
assassination, which we will discuss later. According to Drosnin, the appearance of the ELS for Rabin's name close
to the above phrase about a killer means the prediction of Rabin's
assassination was "encoded" by the Bible's creator by means of the ELS.

The book by Drosnin was soon followed by a
book [3] authored by a psychiatrist Jeffrey Satinover. Strangely, whereas Drosnins's book was repudiated by Rips and some
other adherents of WRR as an invalid vulgarization of WRR's results, Satinover's book met their apparent approval, even though both books
contain very similar examples of ELS arrays, whose interpretation is in no way
supported by any statistical considerations.

Almost immediately after the publication of WRR's paper, some scientists
started expressing skeptical views of that paper. With time, more and more people got involved in the controversy, joining
the dispute on both sides.

The dispute in regard to the authenticity of codes essentially revolves around
three topics, to wit:

1. What is the
statistical probability that the ELS occur in the Bible by chance?

2. Do similar ELS
occur in texts other than the Bible?

3. Are human beings
capable, with or without computers, of creating complex arrays of ELS such as
found in the Bible?

The "code" proponents usually maintain that the probability in question is
extremely small. To the second and third questions the "pro-code" people
usually answer "No."

The numerous examples of the ELS found in the Bible and demonstrated by Witztum,
Drosnin, Satinover, and others have made a strong impression on many people.
Indeed, it seemed hard to imagine that so many coincidences could have happened
by chance. Therefore I decided to first test how often various ELS appear in
randomly chosen texts in various languages.

For my first test I chose several short texts in Russian, English, and Hebrew.
Following the method used by all Bible code searchers, I removed from the tested
texts spaces between the words and all punctuation marks, so the texts converted
into continuous strings of letters. As
soon as this was done, in all the tested examples, numerous individual ELS
popped into sight from the text. As expected, most of those ELS happened to be
three-letter words, but some four-, five-, and six-letter words could be
identified on almost every page as well.

The code proponents often demonstrate "arrays" of semantically related ELS
they find in the Hebrew Bible. I could easily find many similar "arrays" of
semantically related words in the form of ELS in every text I tested. Here is an
example. One of the texts I tested was taken from a short story I wrote in
English many years ago.

Fig.
2 shows the text's segment in
question. Its length is 1460 letters. This segment was chosen without any
preliminary trials, right from the first page of the story.

Among the abundant ELS found in that segment, there were many three-letter
words, fewer four-letter words, still fewer five-letter words, etc. In fig. 2,
only a few of those ELS are marked by straight lines along those ELS, with
arrows indicating the direction of reading. (If the skip equals the width of the page, the ELS appears on a vertical
line. If the skip differs from the page width, the corresponding ELS' appear
on straight lines which are at some angle from the vertical.) In the middle of
the page, there is an ELS for the six-letter word TORVIL (marked 1) situated
vertically with the skip of 68 (since the page's width is 68 letters.) Parallel to it, in the adjacent vertical, there is an ELS for
the word ICE (marked 2) with the same skip of 68. Across the page, there is an
ELS for word DEAN (marked 3) with the skip of 70. Close to both TORVIL and DEAN,
ELS for the word WIN appear twice, one with a skip of -70 (marked 4) and the
other with a skip of -140 (marked 5.) Of
course, Torvil and Dean used to be famous champions in figure ice-skating. The
story in question was written about 10 years before TORVIL and DEAN demonstrated
their skills on ICE, WINning championships twice. Using the "logic" usually applied by Drosnin and other code
proponents, one may suggest that I predicted an event which would occur later,
and had "encoded" my prediction in the form of a combination of several ELS
placed close to each other in the text of my story. I did not.

Among the ELS for 4-letter words in that page were LAND (not far from a
three-letter word SEA), LULL, TILT, ODOR, etc. I discovered similar occurrences
of "arrays" of ELS, seemingly related by meaning, on almost any page of any
texts I tested in Russian, Hebrew or English (many examples can be seen at B-Codes Page). These simple tests have
shown that the phenomenon of ELS is very common, and that many individual ELS
and "arrays" of ELS appear in any text. The reason for that is, of course,
the fact that any language consists of a vast number of words.

I performed the described
non-computerized tests before I came across the Web publications by Dr. Brendan
McKay (In Search of Mathematical Miracles and Dave E. Thomas (Hidden Messages and The Bible Code.)
I was gratified to find that my conclusion turned out to be well in agreement
with the multiple examples of ELS clusters found by McKay in a number of
non-biblical texts, most notably in Moby
Dick, as well as with many examples of similar clusters of ELS in English
texts demonstrated by Thomas (they
used computer programs.)

The proponents of the Bible "codes" claim that the alleged creator of the
"codes" must have possessed superhuman abilities, since neither a human mind
nor the best computers available to us are capable of creating such a complex
web of ELS. I would like to refer
to a book [4] by David Kahn, who is one of the foremost experts in cryptology. In that book one can find a plethora of information about the ability of
men and women to both encode and decode information using methods whose
complexity and sophistication make the alleged Bible codes look simply
primitive.

Indeed, let us recall one of the simplest tools used for encoding secret
information. It was apparently invented by the renowned Italian mathematician
and writer Girolamo Cardano in the 16th century.

The method is as follows. In a sheet of paper, a set of holes is cut forming the
Cardano grille. The grille
is placed over a sheet of blank paper and the message to be enciphered is
written through the holes, one letter per hole. Then the grille is removed, and
the blank spaces between the letters of the secret message are filled with a
text that has some innocent contents. The letters of the encoded message thus
become parts of the overall text, but now are separated by "skips." To
decode the message, a grille identical to that used for encoding is placed over
the text, and the secret message is read through the holes. When the holes are
cut at equal distances, it constitutes the "simple" Cardano grill. It,
of course, produces ELS exactly like those discovered in the text of the Bible.

When I was a kid of about 12, I and a few of my friends used
to send each other secret messages in the classroom right under the nose of our
teacher. To encode a message, we used several techniques, including the Cardano
grille. We had no idea that it was invented in Italy in the 16th century.
Sometimes we used a grille with equidistant holes ("simple" Cardano grille) thus creating sets of ELS not unlike those
found in the Bible. On other occasions we used a grille in which the distance between the holes would either increase
or decrease from letter to letter. It thus produced encoded messages where the
"skip" changed from character
to character. As I will show later in this article, similar "codes" with regularly
increasing or decreasing "skips"
can be easily located in any text, as well as can ELS.

Especially for this article, I decided to try to encode again (like I did as a
12 year old) some simple phrase, being willing to spend on that task no more
than 10 to 15 minutes. First I wrote the following phrase:Rabin Will Die, which, in the
parlance of cryptology, would be my
plaintext.The choice of this particular expression was due to the
widely publicized alleged prediction of Rabin's assassination given in the
already mentioned book by Drosnin. I
wrote the letters of the above expression on a piece of paper, leaving spaces
between letters which would enable me to insert 9 other letters between any two
consecutive letters of the above expression.Then I wrote between the letters of that expression a text, which, even
if it is not very sophisticated (and slightly imperfect grammatically) is
nevertheless meaningful. The entire
exercise took 9 minutes. Here is my
resulting ciphertext: Rivers are dAmningly roBust in some Indian coloNies, moving
Water unerrIngly, thus aLleviating Lust for the Drinks, forcIng people rEconcile
with the otherwise harsh climate. (The
hidden plaintext is shown in
capitals.)

The intended recipient of that message, knowing the skip
(10 in this case), would have no problem in decoding the message. If a serious
need existed for me to prepare a secret message using Cardano grille,
doubtlessly I would be able, by spending more time, to write a much more
sophisticated ciphertext.

The conclusion: a human mind is quite capable of creating arrays of ELS
not unlike those found in the Bible.

An impressive confirmation of the above conclusion was demonstrated by Mr. Gidon
Cohen of York, Great Britain. The Canadian Christian preacher Grant Jeffrey who
propagandized the alleged biblical codes in his publications, was especially
impressed by a segment of a biblical text wherein the names of 25 trees happened
to occur as ELS. In Jeffrey's opinion, such an array of 25 trees' names
"encoded" in the form of ELS within a relatively short text, could
neither happen by chance nor have been created by a human mind. Confident
that the array in question can only be of divine origin, Jeffrey challenged
anybody doubting the reality of the "code" to compile an English
text of comparable length in which the ELS for the names of any 25 trees would
be found. Jeffrey offered one thousand dollars to anybody who will
successfully perform the task. It did not take long until Mr. Gidon Cohen
of York, England, submitted an English text of about 300 words within which he
successfully inserted ELS for the names of 29 trees. (The text in question
can be obtained directly from Mr. Cohen).

By refuting the claims about the necessity
of a superhuman mind for the creation of the alleged codes in the Bible I am not
suggesting that the ELS in the Bible had been created by men or women. The
explanation best compatible with the factual evidence is that arrays of ELS
appear in the Bible not by design but are chance coincidences.

Among the publications about the Bible code there are several books by a
Messianic pastor Yacov Rambsel, and by the already mentioned Jeffrey. For
example, let us look at Jeffrey's book
Signature of God[5]. In this book,
Jeffrey shows six examples of ELS found in the Hebrew Bible, which, in his
opinion, spell word Yeshua, the Hebrew
form for Jesus. The examples of the passages from the Hebrew Bible are printed in
Jeffrey's book, preserving what is called in Hebrew "nekudot," meaning
diacritical marks (dots and short bars) placed either under or above or within
letters (which in the Hebrew alphabet all are consonants) to indicate the
accompanying vowels. The presence of "nekudot" forces a definite reading
for each syllable. Accounting for the printed "nekudot," not a single
example in Jeffrey's book does indeed spell Yeshua, but rather meaningless combinations of letters spelling
non-existing words like Yasvei, Yashaua,
and the like. Of course this observation is only a testimony to Mr. Jeffrey's
ignorance of Hebrew rather than an argument against the existence of the
"codes." Despite the errors by
Jeffrey, the existence of numerous ELS in the Hebrew Bible which spell not only
the word Yeshua, but also combinations
of words like Yeshua Shmi (My name is
Jesus), Yeshua Yakhol (Jesus can), Dam
Yeshua (Blood of Jesus), Yeshua More
(Jesus teacher) etc, can be indeed easily demonstrated. The question is, though, whether such ELS are unique for the Bible or
they occur commonly in all texts.

I decided to search in non-biblical Hebrew texts for ELS spelling word Yeshua and its combinations with some other words, such as those
listed above, all of which have been found by Rambsel in the Bible and touted as
proofs of his beliefs. I randomly
pulled from a shelf a few Hebrew books. The first one happened to be a book by a
contemporary Israeli writer Dahn Ben-Amotz, published in Tel-Aviv in 1979 by
Metziuth Publishers. The title of the book is Ziunim Ze Lo Ha Kol, which translates as Screwing is Not Everything.

In Hebrew, word Yeshua is a
four-letter one (Yud-Shin-Vav-Ayin.) Sometimes
Rambsel used, instead of the four-letter form of Yeshua, the shorter,
three-letter version (Yud-Shin-Ayin.) Likewise, I decided to look for the
three-letter version's occurrences as well.

I leafed randomly through Ben Amotz's book, and soon I located, on page 47,
ELS spelling the two words Yeshua Shmi
(My name is Jesus) one right after the other, both with a skip of only 2. On
page 23, within only three lines of text, ELS for words Dam
and Yeshua, meaning Blood of Jesus,
occurred one after another, both with a skip of only 3. On page 27, within only three lines of text, the same ELS for the words Dam
Yeshua, appeared, both with the same skip of 4. On page 63, within only 2
lines of text, an ELS for the word Yeshua
appeared twice, once in the three-letter version (Yud-Shin-Ayin) with a skip of
3, and once in the four-letter version (Yud-Shin-Vav-Ayin) with a skip of -1. In the same two lines of text the word Moreh
(Mem-Resh-Hey, meaning teacher) appeared 3 times with skips of 3, 4, and -6. In
the same lines the word Mori (Mem-Resh-Yud)
meaning my teacher appeared
with a skip of -5. The characters of the words More
and Mori appeared interspersed
with the characters forming the ELS for Yeshua.
On page 164, within three paragraphs, an ELS for Yeshua appeared 4 times, three times in the three-letter version
(with skips of 2, 4 and -6) and once in the four-letter version, with a skip of
7. On the same page, within the same three paragraphs, the four-letter word Yakhol
(Yud-Khaf-Vav-Lamed) appeared with a skip of 51. The characters of the ELS
for Yakhol were interspersed with those for Yeshua. The phrase Yeshua
Yakhol means Jesus Can or Jesus is Able, and, when found by
Rambsel in the Bible, was interpreted by that writer as one of the proofs of his
claims.

I found in Ben-Amotz's book many more similar groups of ELS spelling every
single phrase Rambsel found in the Bible, which included word Yeshua, I found also, just as easily, many similar phrases including word Yeshua
in a number of other Hebrew books. One of the books in question was the textbook
Geography of the Land of Israel,
published in 1975 in Tel-Aviv by Am Oved publishers. In fig. 3, page 91 from that book is shown.

Fig.3

On the very
first line of that page (right under the section's heading) there is an ELS with a
skip of 6 which reads Tora. In the second line from the top, the third letter from the right is Yud. It starts an ELS with a skip of 4 which reads Yeshua. Right after that ELS, in the same line, there is another ELS, with a skip
of 2, which reads Mori (my teacher). One
line down, starting from letter Mem in the fourth word from the right, there is
an ELS with a skip of 2, which reads More
(teacher). Finally,
in line 8 from the top (not counting the heading) letter number 9 from the left
is Yud. It starts an ELS with a skip of -14 (i.e. from left to right)
which reads Yakhol (can, or is able). Rambsel and Jeffrey touted occurrences of similar ELS in the Bible as miraculous
confirmations of their beliefs.

Is
page 91, shown in Fig,. 3, an exception? In
Fig. 4, page 140 from the same textbook is shown.

There are two groups of ELS on that
page. Let us look at the group of ELS at the top of the page. In the third line
from the top (not counting the heading) the second letter from the right is
Shin. It starts an ELS with a skip
of 3, which reads Shmo (his name).
In the same line, to the left of that ELS, there is another
ELS with a skip of -3, which reads Yeshua. Two lines down, there is one more ELS with a skip of -4, which reads Shmi
(my name). Finally, one more line down, there is an ELS with a skip of -1, which
once again reads Shmi. Of course, if all the above ELS were found in the text of the Bible, then
Rambsel and Jeffrey, and with them their gullible readers, would be announcing,
with delight, a miracle confirming their beliefs.

My conclusion was obvious, namely that a variety of ELS, including combinations
of word Yeshua with various other
words occur quite commonly in non-biblical texts as well as in the Bible.
Therefore their appearance in the Bible does not constitute a proof of anybody's
views or beliefs.

When I was done
with Jeffrey and Rambsel, I learned about some other books also discussing the
Bible code, of which two best known were those by Drosnin and Satinover. Could it be that these writers have done a better job than Jeffrey and
Rambsel and provided some stronger evidence in favor of the alleged code in the
Bible? Let us look at these two best sellers.

Drosnin claims to have made amazing discoveries in the Hebrew
Bible, where many predictions of future events have been allegedly "encoded" as arrays of ELS. The most acclaimed one of those predictions is probably that
of Yitzhak Rabin's assassination. The array of ELS in question, reproduced as a
table in Drosnin's book, comprises the name of the late Israeli Prime Minister
(found with a very large skip of 4772
characters) as well as the phrase Rotzeakh
Asher Irtzakh, meaning killer who will
kill, the sets of letters denoting the year (in the Hebrew calendar) when
Rabin was assassinated, and also the name of the killer, Amir.

Looking at the table in question reveals that of those words only the name of
Yitzhak Rabin is an ELS (with a very large skip.) All the rest of the words in
the set are just parts of the regular text, so are not codes at all. Moreover, the ELS for Rabin's name extends over more than 30,000
letters of the text. Hence, the
concept of "proximity" of all
the mentioned words to the ELS in question is rather ambiguous. The illusion of
proximity has been created by presenting the text of the Bible as a table 4772
letters wide. Otherwise, all
those words allegedly situated close to the ELS in question, would actually be
found scattered over the length of the text.

Since many people remained skeptical in regard to Drosnin's claims, that author,
in an interview, published in Newsweek
(on June 9, 1997) said that if somebody found a prediction of a Prime Minister's assassination in Moby Dick, then he, Drosnin, would
admit being wrong. I assume that by saying Moby
Dick Drosnin actually meant any book other than the Bible.

Not to look too far, I turned again to the book by Ben-Amotz, mentioned earlier
in this article. Now I decided to
look for some ELS related to Rabin. Again, I did not use any computer program,
so I automatically limited myself to relatively short skips and only those ELS
located close to each other. Following Drosnin's practice, I counted only the
letters, ignoring the spaces between the words, commas, periods, etc.

The word Rabin (four Hebrew letters,RESH-BETH-YUD-NUN) popped up on page 33, in the uppermost paragraph on that
page, with a skip of -35. Furthermore, within the two uppermost paragraphs, consisting of only
about 600 characters, the following ELS appeared, one following the other: 1) An
ELS consisting of four Hebrew letters, with a skip of only -3,
YUD-RESH-TZADE-KHET, which reads Irtzakh
(meaning he will kill). 2) Three Hebrew characters, with a skip of -15,
RESH, HEY, MEM, which is the Hebrew abbreviation for Rosh Ha Memshala, meaning Prime Minister (this abbreviation is commonly used in Hebrew, and
was also used as such by Drosnin.) 3) The following Hebrew characters with a
skip of -8, GIMEL, BET, RESH, which is normally read as Gever (meaning Man)
but is interpreted, for example, by Satinover as Gibor (meaning Hero or Mighty
man). 4) As mentioned before, the word Rabin with a skip of -35. So, in just two short paragraphs in a randomly opened page, in a randomly
chosen Hebrew book, without employing any rearrangement of the text like that
usually performed by Drosnin, a set of 4 ELS, all words appearing in one chain, in a very close proximity, reads: he
will kill, Prime Minister, hero, Rabin. If we wished to calculate the
probability of the occurrence of these four ELS within those short paragraphs,
using methods of calculation routinely employed by the code proponents, we would
arrive at an astonishing number of about 1 in several billions. (As is shown at Improbable probabilities,
as well as in the next section of this article, the calculation of
probabilities performed by the code proponents actually provide meaningless
numbers.) What seemed to be still
missing in those two paragraphs, was an ELS for the name of Rabin's killer,
Amir. So I recalled my experience
with Cardano grille, where I and my
12-year old friends sometimes used a grille with gradually increasing or decreasing skips. Indeed, if the
alleged creator of codes hid information in the form of ELS, couldn't he/she
use also, for example, DSLS (Decreasing Skip Letter Sequences) as I and my
friends did at the age of 12? It
would be even a better code, wouldn't it?

I looked for any occurrence of such a "code." In the same chain of words
mentioned above I found the following 4 Hebrew letters: AYIN, MEM, YUD, RESH,
which read Amir, with the skip between
the first and the second letter being -8,
between the second and the third letter being -9, and between the third and the fourth letter being -10. What a nice
regularity, and what a beautiful code, isn't it? Now all "encoded"
words, all within the same two paragraphs, with small skips, in a chain, read Amir Will Kill Prime Minister Hero Rabin.

Following Drosnin, one may say, "What an amazing discovery!" Obviously, the
probability of those words appearing within two short paragraphs, in such a
close proximity to each other, must be exceedingly small. Then shall we conclude that in a book published in 1979 a message was
deliberately encoded predicting the
assassination or prime Minister, the hero, Rabin, by Amir, some 16 years
before it actually happened?

The above example from Ben-Amotz's book in regard to Rabin's assassination
was posted on the Internet in January of 1998. So far, despite his promise, Drosnin did not reply in any form.

While I found the "prediction" of Rabin's assassination in a Hebrew book,
what about Moby Dick, specifically
mentioned by Drosnin in his interview?

Responding to Drosnin's challenge, an Australian mathematician and computer
expert, Dr. Brendan McKay, applied a computer program searching for ELS to the
text of Moby Dick. He discovered there numerous arrays of ELS "predicting" assassinations of many prime ministers and other known
personalities, including Chancellor of Austria Dolphus, Leon Trotsky, Indira
Gandhi, Abraham Lincoln, Martin Luther King, and, of course, Itzhak Rabin.

The book by Satinover also contains similar examples of alleged amazing finds in
the Bible. Although Satinover uses an additional criterion of nearly minimal skips, his finds still are almost certainly results
of a random chance.

Here is one example. In chapter 10 of his book Satinover presents a number of
"arrays" of ELS found in the Hebrew Bible by himself as well as by
Witztum, by Rips, and by Moshe Katz. These arrays supposedly contain encoded
information about Satinover's ancestor, Rabbi Abraham nicknamed The Angel, as well as about Emperor Franz Joseph of Austria, about
diabetes, about the AIDS epidemic, and about the assassination of Anwar Sadat.

I decided to look for similar "arrays" in a text other than the Bible. I
chose the array from page 164 of Satinover's book, attributed to Witztum. It
deals with the AIDS epidemic. The array in question contains ELS which spell the
following words: AIDS (Aleph-Yud-Dalet-Samekh),
Mavet (Mem-Vav-Tet, meaning death),
BeDam (Bet-Dalet-Mem, meaning
in blood), The HIV (Hey-Hey-Yud-Vav)
etc.

Again, I leafed randomly through the book by Ben-Amotz, mentioned before. On page 67 I
found the following ELS, all situated within the two uppermost paragraphs, which
contained a total of about 600 characters: BeDam (Bet-Dalet-Mem, meaning "in blood") with the skip of 15. Mavet
(Mem-Vav-Tet, meaning "death") with the skip of 10. HIV (Hey-Yud-Vav) with a
skip of 18. Additionally, in the same paragraphs there was word Kholi (Khet-Lamed-Yud) meaning
"disease."

There seemed to be no ELS for AIDS in
that short text. Again, remembering the Cardano grilles I used as a kid and
realizing that the "code" might not necessarily be limited to ELS, I
looked for word AIDS (Aleph-Yud-Vav-Samekh)
"encoded" with a regularly increasing or decreasing skip. I found that
word in the same paragraphs, the skips being -18 between the first and the
second letter, -17 between the
second and the third one, and -16
between the third and the fourth letters. How small is the probability of the
described occurrences expected to be?

I easily found a very similar group of ELS, again without using a
computer, in the above mentioned textbook "Geography of the Land of Israel."
Let us look again at page 140 of that book, shown in Fig. 4. This time look at
the group of ELS in the lower part of that page. In the sixth line from the bottom, the seventh letter from the right is
Khet. It starts an ELS with a skip
of 4, which reads Kholi (disease.) One
line down, the first letter from the right is Hey. It starts an ELS with a skip of 3, which reads HIV. In the same line,
to the left from the above ELS, there is an ELS with a skip of 3, which reads BeDam
(in the blood.) One more line
down, in the third word from the left, there is letter Mem. It starts an ELS
with a skip 9, which reads Mavet (death.) Is there any difference between these ELS and those found in
the Bible? None whatsoever.

Many proponents of the "code" tried to calculate the probabilities of the
appearance, by chance, of various ELS in the text of the Bible. Unfortunately,
most of these calculations are characterized by errors and misconceptions and
produce unreliable values of probabilities. In such calculations the very small
probabilities (like "one in quadrillions" in Jeffrey's book) are usually
obtained by multiplying many not very small probabilities. For example, the
probability of a particular letter appearing at a certain location in the text
is calculated by dividing the total number of occurrences of that letter in the
text, by the overall text's
length expressed in the number of all letters in it. Even this first step is actually wrong as it only applies to a perfectly
random conglomerate of letters. Meaningful
texts are very far from being random. On
the contrary, the meaningful texts possess a high degree of order in various
forms (see, for example, several articles by myself and B. McKay at A study of certain statistical properties of meaningful texts as compared to randomized conglomerates of letters). Therefore the probability
of a particular letter appearing at a certain site in a meaningful text may be
very different from the probability calculated for a random text.

However, there is a much more serious error in the calculations in question. Having calculated (erroneously, as explained above) the
probabilities for various letters occurring at various locations in the text,
the code proponents then usually multiply
the probabilities found for each letter in the ELS in question and thus arrive
at extremely low numbers. Multiplication
of individual probabilities is routinely employed in the theory of probability,
but the necessary condition to make this multiplication valid is independence
of the multiplied probabilities. If
the individual probabilities are not independent, probability theory treats them
in a different way, as the so called "conditional probabilities." To determine whether the individual probabilities are independent or not
is by no means a trivial task. In
particular, the individual probabilities multiplied by the code proponents are
definitely not independent (see Improbable Probabilities). Therefore the extremely small probabilities of the occurrence of that or
this ELS, or of ELS arrays, which are so triumphantly claimed by the code
proponents like Jeffrey, Rambsel, Drosnin, Satinover, etc, are meaningless
numbers.

Now, let us consider the question: what if the probabilities in question are
indeed very small? Does it mean the occurrence of corresponding ELS must be
attributed to a conscious design?

As is shown in my article at Improbable probabilities, probability theory can not and does not predict whether an event will or will
not occur. Even if the calculated probability of an event is very small, it does
not at all mean that the event in question will not happen. All the probability
theory can assert is that in a very large number of tests the average number of
occurrences of a specific outcome will approach the calculated value of
probability for this outcome if the number of tests increases. No more and no less than that.

The chance of finding in the Bible a particular ELS may be small (but still much
larger than the one usually claimed by the code proponents) but the chance of
finding some ELS, for example combining the word Yeshua with certain other words, is much larger. The reason for that
is that the Hebrew language (and any other language as well) contains so many
different expressions and phrasal constructions that the chance of coming across some seemingly meaningful combination of Yeshua with some
other words in the Torah is quite large. The
fact that many expressions found by the code proponents in the Bible were also
located in non-Biblical texts may serve as a confirmation that calculations of
probabilities, like those often used by the code proponents, are meaningless and
cannot be used to justify any conclusions about the reality of "codes"
in the Bible.

When I felt confident that I was done with such Bible
"code" epigones like Drosnin, Satinover, Jeffrey etc, I turned to the paper
[1] by WRR themselves.

All the Bible code proponents and
defenders refer to that paper with the utmost esteem and repeat time and time
again that WRR's paper had "scientifically proven" the genuineness of the
"Bible code." Among the
arguments in favor of the "codes" authenticity, the WRR's epigones
usually emphasize that Witztum, Rips, and Rosenberg are genuine scientists, that
their paper was published in a prestigious refereed magazine, and that no
skeptical scientists have yet found any errors in WRR's calculations.

All the above assertions are without merit.

First, out of the three authors only Rips is indeed a scientist, a highly
qualified mathematician in the field of group theory (but, unlike many of the
code opponents, not in mathematical statistics). Rosenberg developed a computer program used by WRR. Many similar programs
have been offered, some for free and many more for sale, and none of those
programs' creators (such as Kevin Acre, Dave Thomas, Randall Ingermanson etc.,
all of whom maintain web sites) pretends that their programs qualify as a
scientific achievement. Witztum,
who has characterized himself, in an article published in the Jewish
Action magazine (March 1988) as "the foremost code researcher in the
world" has no scientific credentials (although he has a Master degree in
physics).

WRR's followers have a habit of constantly exaggerating scientific
qualifications of anybody who is or could possibly be in favor of WRR's work. The epithets such as "world-class" mathematician, probabilist, etc.,
are routinely applied to whoever might, often just in passing and often in
ambiguous terms, say anything that WRR's defenders choose to construe as a
support of WRR's conclusions.

Of course, the question of whether WRR are genuine scientists or amateurs is
irrelevant and I have touched on it only as a response to WRR's followers'
persistent misuse of that argument.

Furthermore, if one wishes to use the argument based on the credentials of WRR,
then one has to account for the opinion of many real experts in mathematical
statistics who signed a letter (at Mathematicians' Statement on the Bible Codes) stating that they had personally analyzed WRR's paper and categorically rejected
WRR's conclusions. At the time of the writing of this book more than 50
mathematicians have signed the letter in question. Only experts in mathematical
statistics were offered a chance to sign the letter in question. Were it offered
also to physicists, linguists, Bible scholars, etc, the number of signatures
would certainly be much larger. There
is no document signed by any number of scientists who would support WRR's
conclusions. It can be therefore
stated that the scientific community, and more specifically mathematicians and
specialists in statistical science, have overwhelmingly rejected WRR's work.

To thoroughly analyze WRR's work and to determine where WRR were at fault took
a considerable time. A number of
Internet postings appeared since 1994 in which various facets of WRR's work
have been criticized from various viewpoints. Therefore the WRR's followers'
argument that no scientist could find errors in WRR's paper, is a blatant
distortion of the actual situation. For example, let us mention just one printed
article [7]. Postings on the Internet by scientists criticizing WRR are numerous
(see for example In Search of Mathematical Miracles
or B-Codes Page, both containing many links to relevant websites.) Finally, in May 1999, in the
same Statistical Science journal, a
paper [8] by three mathematicians
and one psychologist (to be referred to as MBBK) offered the most comprehensive
refutation of WRR's
work. The same Dr. Kass who was the
editor of the journal when WRR's paper was printed, supplied a comment to
MBBK's paper, stating that the puzzle offered in 1994 has been successfully
solved and the allegedly amazing results of WRR's experiment have been given a convincing rational explanation.

While the excellent paper by MBBK provides a devastating criticism of WRR's
procedures and conclusions, it does not encompass all
facets of the controversy. In particular, MBBK actually accept some rules of
the game employed by WRR, concentrating mainly on the demolition of the
statistical evidence provided by WRR, but neither rejecting the fundamental
hypothesis by WRR nor analyzing some of the details of WRR's basic approach to the statistical procedures. While MBBK's
argumentation is more than sufficient to completely destroy the credibility of
WRR's conclusions, a look at some other fundamental weaknesses of WRR's work
is also of interest.

In the following sections I will provide both a brief report on the MBBK
formidable analysis of WRR's work and an equally brief consideration of some
additional aspects of WRR's basic approach to the problem in question.

At the beginning of their paper WRR offer an example. If one is dealing with any meaningful text, then, as WRR suggest, it is reasonable to expect that words which are semantically related, for
example, hammer and anvil, occur in the text in close
proximity to each other. Likewise, they contend, if the ELS in the Bible
constitute a "code" inserted there by a superhuman mind, then the ELS which
are related by contents are also expected to occur in close proximity to each
other.

WRR's hypothesis is doubtful even in regard to human-created meaningful texts.
It is easy to imagine a text where the word hammer
occurs very frequently whereas the word anvil
is completely absent. However, for
the sake of argument, let us accept the above hypothesis for human-created
texts. WRR's extension of that
hypothesis to the occurrences of semantically related ELS in the Bible is a jump
across a logical pit. If the "code" has indeed been created by a superhuman mind, then, by definition, we have no
way to know how that mind works. What
is the reason to assume that the alleged superhuman creator of the "code"
wished to place the ELS "related
by meaning" close to each other?

All ELS reported by WRR and their followers are just individual words without
any grammatical connections between them. Despite
a number of attempts, so far neither WRR nor any of their followers succeeded to
prove that sets of ELS' in the Bible form meaningful phrases not to mention
whole sentences. Then what could be
the reason to haphazardly place various ELS in a text "close" to each other? Moreover, it seems more logical (or, better, no more illogical) to guess
that the alleged creator of the "code" would place related ELS as far from
each other as the text would allow. Indeed,
as it was shown in one of the preceding sections, the human mind is quite
capable of creating ELS in any texts. This task is relatively easy if the skips
are short. If the skip exceeds the length of one page, the creation of an ELS
becomes difficult. Hence, if the
alleged superhuman creator of codes wished to "encode" some information
using ELS, it is equally plausible (or, rather, no more implausible) to
hypothesize that "he/she" would rather opt for placing parts of the code at
long distances from each other thus making clear that "his/her" code is not
of human origin.

The hypothesis by WRR which is at the core of their entire effort, seems to be
arbitrary, as it has no foundation in any factual evidence and is not supported
by any argumentation. In other
words, even if WRR managed to prove that semantically related pairs of ELS
indeed occur in the Book of Genesis in an unusually close proximity, this in
itself cannot serve as a proof of the superhuman origin of those ELS.

The proximity of ELS may have quite non-miraculous origin stemming from the
natural properties of texts. Indeed,
research conducted by myself and Dr. B. McKay (see for example our postings at A study of certain statistical properties of meaningful texts as compared to randomized conglomerates of letters revealed some specific pattern of strong organization in all meaningful texts
which are absent in randomized texts. Because
of the high level of order in meaningful texts, every mathematical function
reflecting properties of a text has nearly extreme values for meaningful texts
as compared, for example, with most randomized texts. The complex mathematical quantity WRR use as a measure of
"distance" between ELS is no exception. Hence, even if the unusually close
proximity of semantically related ELS in the Book of Genesis were proven, it
would not constitute a proof of a
supernatural origin of those ELS. Moreover,
as we will see, the unusually "close proximity" claimed by WRR has by no
means been proven in their work.

To verify the results observed for the Book of Genesis,
WRR also conducted their experiment on six control texts. Four of those control
texts had been obtained by permutations of various elements (words, verses, etc)
of the Genesis text. One more control text was the Book of Isaiah, and one more
was the Hebrew translation of Leo Tolstoy's novel War
and Peace.

It is of interest
that among all the texts explored by WRR the worst results (that is, the highest
rank of the "correct" appellations/dates list among all the competing
shuffled lists) was observed for the Book of Isaiah. In other words, the experiment which, as WRR maintain, was successful in
the Book of Genesis was a complete failure in the Book of Isaiah.

Since
many arrays of ELS found in the Book of Isaiah are often demonstrated by the
"code" proponents, it seems
puzzling that WRR did not provide any discussion of their failure to get a
positive result in the Book of Isaiah.

Moreover, how can the complete absence in
WRR's work of any reports on attempts to apply their method to the other four
books of the Torah be explained?

Perhaps the answer can be inferred from the results obtained by some scientists
who filled the void left by WRR. McKay
used the computer program supplied by WRR, and the same list of the rabbis'
appellations/dates as used by WRR, and tried it on the other four books of the
Pentateuch (see the results of that experiment as reported by Dr. A.M. Hasofer
in A Statistical Critique of the Witztum et al Paper). The results were decidedly negative. While
the "correct" list of rabbi's appellations/dates used by WRR, in
conjunction with their method, produced low ranks among one million shuffled
lists in the Book of Genesis, in the other four books of the Pentateuch those
ranks were very high (hundreds of thousands.)

The Judaic tradition holds that the Torah was given on Mount Sinai to Moses as a
string of letters without divisions into words, chapters, or books. The division was performed later. The
boundaries between the five books of the Torah have no real significance and
have been disputed by some prominent figures in Jewish history.

WRR avoid any discussion of why their experiment produced such good results in
Genesis but utterly failed in the other four books of the Torah. The natural
guess which immediately comes to mind is that the list of appellations/dates was
somehow optimized in WRR's study toward the positive outcome of their
experiment with Genesis. As we will see later, this guess finds some
confirmation when a detailed study of the above list is performed. This statement in no way implies that WRR deliberately adjusted their
data to produce the desired effect. The
history of science knows many examples of perfectly honest and unbiased
scientists inadvertently selecting from their data sets those subsets which
produced results meeting the experimenters' expectations and ignoring other
subsets which might produce an opposite outcome.

A rebuttal of the WRR's work from the standpoint of mathematical statistics
and probability theory has been proposed by the above mentioned prominent expert
in mathematical statistics, Dr. A. M. Hasofer. The paper by Hasofer is available
at A Statistical Critique of the Witztum et al Paper. I will briefly present here the
main points of Hasofer's
comments. This section will
necessarily be more technical, as it deals with subtle elements of statistical
procedure. The readers who are uncomfortable with mathematics, can just skip it,
even though it demonstrates serious deficiencies in WRR's statistical manipulation of ELS in
the Bible. '

The part of mathematical statistics which is relevant to the discussion of
WRR's work is "hypotheses testing." The
proper procedure in testing hypotheses from the standpoint of statistics
includes several necessary steps, as follows:

1) The first step is setting the so called "null hypothesis." Usually the null hypothesis is a certain statement implying that, within
the framework of a set of conditions, there is no statistically significant
new phenomenon to be discovered. In
other words, the null hypothesis reflects the expectations that a test aimed at
the discovery of a new phenomenon will fail. In the case of WRR's work, the null hypothesis, in broad terms, is that
the proximity of pairs of semantically
related ELS in the Book of Genesis is such as would be expected if the
distribution of those ELS in the text were purely random. When applying a statistical test, the null hypothesis normally is
expressed in terms of certain quantities to be measured, whose values would
indicate whether the null hypothesis must be accepted or rejected. In WRR's work, such quantities are ranks
of the "correct" list of appellations/dates among one million competing
scrambled lists. The lower are the
ranks in question, the more likely it is that the null hypothesis is wrong and
has to be rejected. As we will see,
the legitimacy of the null hypothesis by WRR, is, at best, uncertain.

2) The next necessary step is setting the so-called alternative
hypothesis. The alternative
hypothesis implies that a new statistically significant phenomenon is likely to
exist. The hypotheses testing
essentially boils down to the competition between the null hypothesis and the
alternative hypothesis, to determine which is more likely to be true. Unfortunately, contrary to the accepted statistical procedure, WRR never
expressly formulated any alternative hypothesis, leaving it to the readers to
figure it out.

3) The next step in a proper statistical test is to define the so-called critical
region and the power of test. The critical region means choosing the boundaries for the set of the
quantities to be measured such that if the measured quantities are found within
those boundaries, the alternative hypothesis is considered as more likely than
the null hypothesis. The power of
test is a quantity related to the size of the critical region. WRR, contrary to
proper statistical procedure, never defined the critical region and the power of
test, thus rendering their conclusions statistically ambiguous.

In particular, trying to interpret what WRR meant by their hidden alternative
hypothesis one comes to the conclusion that the power of test implied in their
study must be 1. In plain words, it means that to produce sufficient statistical
evidence in favor of their alternative hypothesis, the rank of the correct list
of appellations/dates must always be 1, which of course almost never happened in
their tests.

Besides the above deviations from proper statistical procedure, WRR's work
suffers from many other features contrary to the rules of mathematical
statistics.

One of such features is the nature of WRR's null hypothesis. As McKay et
al have indicated in their
paper [8], the null hypothesis adopted by WRR actually reflects not only the
properties of the text of Genesis, but also those of the appellations/dates
list. The ranks of the
"correct" list are partially determined by the composition of the list and
not only by the text of Genesis. Indeed,
as McKay et al demonstrated, slight
variations in the list's composition may result in drastic changes of ranks. I
will return to that observation a little later.

Another serious deficiency in WRR's statistical
treatment, pointed out both by Hasofer
and McKay et al, is the asymmetry
between the scrambled lists. This
asymmetry is created by the following circumstances. For each "famous rabbi" WRR use several appellations (up to 11
appellations per person.) They also use several versions of dates of the
rabbis' deaths/births. A total of
298 combinations of names with dates existed for WRR's list of rabbis. 135 of those combinations are not
found as pairs of ELS in the Book of Genesis. WRR used the other 163
combinations. Let us assume that rabbi
A has three appellations and two dates in the "correct" list. (For example,
the dates can be one of birth and the other of death, and appellations can be in
several forms, one being only the last name, the other a nickname, one more the
name plus the title, etc.) Each combination of an appellation with a date
provides an entry to the list of appellations/dates. Then rabbi A contributes
six entries to the "correct" appellations/dates list. Let us assume that rabbi B has only two appellations and one date in that
list. Hence, rabbi B contributes two entries. Then these two rabbis contribute
eight entries to the "correct" list. There is, among the mismatched lists,
one where the appellations for rabbi A are combined with the dates for rabbi B,
and vice versa. Then rabbi A would contribute not six but only three entries to
the scrambled list, and rabbi B, instead of two, would contribute three entries. Hence the two rabbis in question would contribute to the scrambled list
the total of six entries instead of eight in the "correct" list. Different
mismatched lists will have different numbers of entries, and this will
drastically affect the values of aggregate criteria of "proximity." The symmetry of sets of numbers to be compared is a necessary condition
for a proper statistical test, and such symmetry is absent in WRR's work.

One more item in WRR's work which met with criticism was their choice of the
measure of the distance between the ELS. That
distance has been defined by WRR using a very complicated procedure. In order to calculate that distance, in some cases as many as
over 6 million arithmetic operations are needed. In one of Hasofer's example, the "distance," according to WRR's formula,
between two ELS that touched each other, turned
out to be much larger than the "distance" between two other ELS which were
quite obviously remote from each other. Since the aggregate criteria of
"proximity" and therefore the ranks of the "correct" lists were calculated by WRR based on
their "distance," I believe this Hasofer's observation alone renders all WRR's
conclusions meaningless.

So far our discussion of WRR's paper was all within the framework of
statistics. This consideration, based mainly on the observations by
experts in mathematical statistics, among them Dr. Hasofer and Dr. B. McKay et
al, has shown that the results of WRR's statistical treatment of their
experimental data are unreliable and cannot serve as foundation for their
extraordinary claims. Now, however, for the sake of discussion, let us assume
that WRR's claims are based on an impeccable statistical procedure, and that,
therefore, their null hypothesis has to be rejected and their (not explicitly
expressed) alternative hypothesis has to be accepted. From the purely statistical viewpoint this would be the satisfactory
outcome of their experiment.

In this section I will offer some considerations which will lead us beyond the
purely statistical procedure.

In any text on mathematical statistics (for example [9]) there are warnings
against overestimation of the validity of statistical analysis. Acceptance of the alternative hypothesis completes the statistical
analysis, but it is not necessarily the ultimate test of the problem in
question. The only firm conclusion
is that under the defined set of
conditions, the alternative hypothesis is more
likely than the null hypothesis. There always exists a probability that some
other hypothesis, not formulated in the study, is even more likely than the
formulated alternative hypothesis.

There is a difference between a statistical hypothesis and a general scientific
hypothesis. A statistical hypothesis necessarily deals with random variables. Having "proven" the alternative hypothesis
does not necessarily mean that it has been proven as the general scientific
hypothesis.

Consider a trivial example. Imagine
that a study has been conducted in which the frequency of cases of tuberculosis
was measured in a certain country and in the course of that study it has been
noticed that the cases of tuberculosis seemed to be less frequent among the
people owning gold watches. To verify this
observation, a systematic statistical research has been initiated. The null
hypothesis was that the frequencies of TB and of gold watches' ownership are
distributed in a random fashion and are independent of each other. The
alternative hypothesis was that the cases of TB are negatively correlated with
gold watches' ownership, since presumably rich people both more often possess
gold watches and, because of better nutrition and living conditions, less often
have TB. For obvious reasons,
in the properly conducted statistical study, the alternative hypothesis would
overwhelmingly win. From the purely
statistical viewpoint, the strong negative correlation between TB and gold
watches would be well established. As
a general scientific hypothesis, this conclusion will be meaningless and no
statistician would recommend curing TB by distributing gold watches.

Unfortunately, despite many similar well known examples, some statisticians
develop a mind set that makes them sometimes lose the perspective of the general
scientific hypothesis and view the statistical proof of an alternative
hypothesis as the end in itself.

In the above example, the meaninglessness of the statistical result becomes obvious if some material evidence is added to the study. Namely, attempts to cure tuberculosis by distributing gold
watches, whose results can easily be foreseen, will rebut the statistically
valid, but nevertheless meaningless results. In some other cases, the material
evidence may be not readily available, but the results of a statistical study
can still be judged to be meaningless just by analyzing the behavior of
quantities employed to characterize the null- and the alternative hypothesis. This was, in my opinion, the case with WRR.

To explain what I mean, consider an example.

Imagine that a study is being conducted to compare the vitamin C concentration
in apples vs. oranges. The
amount of the vitamin in each individual apple or orange is different, therefore
this quantity is a typical random variable and hence the legitimate object of a
statistical study. It is impossible to analyze every apple and orange, so a
selection is made of, say, 100,000 apples and the same number of oranges from
various crops and regions. The
amount of the vitamin is then measured in each selected apple and orange. Then it is necessary to choose a certain aggregate measure
characterizing the statistically averaged amount of the vitamin in each of the
two types of fruit. Assume that in order to subject the study to a self-control,
two different aggregate measures are chosen. For example, one measure can be the
arithmetic mean of all the measurements, (which will be denoted P1 )
while the other aggregate measure (denoted P2) can be, for example, a
product of data for all individual examples, excluding the data within the
lowest five percent and the upper five percent of the measured vitamin's
concentrations. Obviously, P1
and P2 will be different numbers, but that will not by itself
invalidate the results. What, though, if the arithmetic mean P1 will
be found larger for apples, while the product P2 will be found larger
for oranges? In other words, if we
rely on P1 , then the rank
assigned to apples will be 1 and to oranges, 2. However, if we rely instead on P2,
the oranges will be assigned rank 1, and the apples, rank 2. Hence according to
P1, we conclude that
apples contain, on the average, more vitamin C than oranges, but, according to P2,
we conclude that apples contain, on the average, less vitamin C than oranges. The two aggregate
criteria produced mutually exclusive results. The only possible conclusion is
that at least one of the two aggregate criteria, and perhaps both of them are
unreliable.

That situation is exactly what happened in WRR's research. They suggested two formulas, one for calculating what they denoted
statistic P1 and the other
for statistic P2, both
being the supposed cumulative measures of the "proximity" of pairs of ELS in
a text. Before WRR published their
paper in Statistical Science, they made public some tentative reports on their study, in
which P1 and P2
already appeared and were claimed by WRR to represent certain probabilities.
P1 was introduced as the
probability that the number of those inter-ELS distances for rabbis'
appellations and dates, which is not more than 0.2 (on a scale between 0 and 1)
equals a certain number m.P2 was first
introduced as the probability that the product of all inter-ELS distances for
rabbis' appellations and dates is not larger than a certain number X. However, when the journal's referee (Dr. P. Diaconis) had indicated
that P1and P2did not meet the requirements for probabilities calculation, WRR changed
their definition saying instead that P1
and P2 are not
probabilities but just certain cumulative quantities reflecting, in some not
precisely determined manner, the overall proximity of semantically related ELS. Later WRR added two more aggregate measures of proximity denoted P3
and P4, calculated by the
same formulas as P1and
P2 respectively but applied
to somehow modified lists of rabbis' names. The ultimate results reported by
WRR were not the values of the four cumulative measures per
se, but rather the ranks of the
correct list among one million of competing scrambled lists. What is interesting in those data is the different ranks of the "correct" list of appellations/dates,
depending on which of four P was used.

For example, in an experiment reported by E. Rips to the Israeli Academy of Sciences in 1996 (it can be viewed at Hidden Codes in Equidistant Letter Sequences in the Book of Genesis) the rank of the "correct"
list was 14 if statistic P1was used but 2724 if P2was used. If the rank of the "correct" list is 14 it means that among
the 999999 competing scrambled
lists there are 13 lists where the proximity of ELS for rabbis' appellations
to those of the corresponding dates is better
than it is in the correct, unscrambled list. If though the rank of the correct
list is 2724, it means that there are not 13 but 2723 scrambled lists with a
better proximity than in the correct list. Then at least 2710 scrambled lists are believed to have a better
proximity than the correct list, if we rely on criterion P1, but the same at least 2710 scrambled list are believed to have a worse
proximity than the correct list if we trust P2. Which result do we believe?

The only possible conclusion is that either P1 or P2 or
perhaps both, are unreliable measures of "proximity." The same relates toP3 and P4.

In WRR's experiments, the ranks of the "correct" list of
appellations/dates were consistently low (even if not equal) for all of four P
(mostly between 4 and several thousand, out of one million of competing lists.)
Hence, from the purely statistical viewpoint (and ignoring the above discussed
deficiencies of WRR's statistical treatment) there are sufficient grounds to
reject the null hypothesis and to accept the (not defined) alternative
hypothesis, as the more likely one. That is what most of the critics of WRR's
work do, concentrating instead on the weaknesses of WRR's data lists (which we
will discuss in the next section). However,
even if we were to accept WRR's conclusion as a legitimate result
of their statistical study, that conclusion would still remain an unsubstantiated hypothesis under the requirements
of a general scientific hypothesis. One of the reasons for that is the
contradiction between their four cumulative measures of proximity. The last
consideration is, of course, of purely academic significance, because, as was
shown in the preceding sections, the statistical study by WRR is itself fraught
by irregularities.

All that said, one more question remains, namely how to specifically explain the
consistently low (even if not equal) ranks of the correct appellations/dates
list WRR obtained for all four P. We
will now discuss this question, which has been decisively answered by several
scientists, including Barry Simon and most of all by McKay et
al.

A prominent mathematician, Dr. Barry
Simon, the IBM Professor of mathematics and theoretical physics at Caltech, the
author of many books and hundreds of scientific papers and a recipient of many
awards, is just one of those many scientists who have unequivocally rebutted
WRR's method and conclusions. In order to explain the paradoxical results of
WRR's statistical study, Simon suggested [10,11] the concept of "wiggle room." This term relates to the large number of choices WRR
could make when compiling their lists of rabbis' appellations and dates. For
example, WRR inform us that the data on the famous rabbis' appellations and
pertinent dates had been extracted from the Responsa
database maintained by Bar-Ilan University. An examination of the Responsa database revealed that the 66 rabbis included in the two
WRR's experiments had been known by at least twice as many names and
appellations as WRR used in their lists. WRR
seem to have made a number of arbitrary choices about which appellations and
which dates to include or not to include in their list.

The natural question is, of course, whether or not the described "wiggle
room," which was available to WRR, was sufficient to produce a
"successful" list, and whether or not other choices available to WRR, if
incorporated in their list, would invalidate their results (i.e. make the ranks
of the correct list large enough to deprive them of statistical significance.)

The decisive answer to that question has been given mainly by B. McKay et
al in a number of Internet postings and in their paper [8].

McKay et al have shown how small variations in the list compositions can
drastically change the rank of that
list among its scrambled versions. In
one such example, Witztum claimed to obtain rank of 1 for a list of data where
he juxtaposed ELS for a series of words related to the dreadful Nazi
extermination camp at Auschwitz. McKay
used a data list which differed from that used by Witztum only in minor details,
and was at least equally legitimate.
For that, slightly modified list, the rank was found to be 289,000 instead of 1.

In regard to the "famous rabbis" experiment, McKay and Bar-Natan similarly
slightly modified the list of appellations/names. They preserved 80% of WRR's list, deleted a few names whose inclusion
in WRR's list rested on a shaky foundation, and added a few names whose
inclusion had at least as good reasons as all the other names in WRR's list. Using this list of names/dates, which was justified at least to the same
extent as that of WRR, McKay and Bar-Natan found no statistical evidence
whatsoever of unusual proximity of the pertinent ELS in the Book of Genesis.

Analyzing WRR's work MBBK found, for example, that if only four (out of 32)
rabbis whose inclusion supplies the strongest contribution to WRR's low rank
for the "correct" list of appellations/dates are deleted, the statistical
significance of WRR's
result changes from 1 in 62,000 to 1 in 30. Deletion of only one appellation (out of 102) changes the significance
level by an order of magnitude. Deletion of only five names results in the
change of significance level by three orders of magnitude.

In another experiment, McKay and Bar-Natan demonstrated how easily the list of
names/dates can be "cooked" to achieve any desirable outcome of the
experiment. Their slightly "cooked" list produced no significant result in
Genesis, but produced what looked like statistically significant results in the
Hebrew translation of Tolstoy's War and
Peace.

In their paper [8] MBBK have shown that the "wiggle room" available to WRR
was more than sufficient to compile a list of names/dates which would produce
low values of rank in any selected text. Of
course, the question of whether WRR made their choices deliberately or whether
these choices were the result of an inadvertent optimization, is irrelevant. There is no evidence and no reason to suspect WRR in a
deliberate manipulation of the data, but the existence of a wiggle room, in the
traditions of scientific discourse, is in itself a sufficient reason to subject
WRR's claims to serious doubts.

WRR's claim is, in particular, based on their assertion that the text of the
Book of Genesis has been preserved intact, letter by letter, since it was given
to Moses on Mount Sinai some three thousand years ago. Indeed, if the ELS in the Bible constitute a God-inserted
"code," this code must have been present there from the very beginning. Any, even small, changes in the text would necessarily damage the code,
which depends on the exact sequence of all letters.

The claim by code proponents in regard to the supposed perfect preservation of
the Torah text has been convincingly refuted, for example, by Dr. Jeffrey Tigay, who is a Professor of Hebrew and Semitic languages
and Literature at the University of Pennsylvania (the article by Tigay can be
viewed at In Search of Mathematical Miracles,
McKay's
web site) as well as by Menachem Cohen who is a professor of biblical studies in
Israel as well as the editor of authoritative editions of the Bible's text.(Cohen's article can be seen at my website B-Codes Page) A similar criticism has been offered also by MBBK. It has been established that the text of the Bible had undergone many
changes in the course of its existence.

The code proponents sometimes respond with two different explanations. One is
that the codes found in the Bible are just remnants of the original God-designed
codes. To that argument, MBBK
replied that deletion even of one letter in a thousand must completely destroy
any ELS-based code. Given the extent of changes the text of the Bible must have
experienced, there is no chance that any original "code" could have
survived. MBBK have supported that
claim by experimental data. In one experiment, MBBK deleted only 50 letters from
the text, and this resulted in the complete disappearance of the effect claimed
by WRR.

Another explanation by the code proponents is that God knew in advance which
changes would occur in the text of the Bible and adjusted the sequences of
letters in it in such a way as to ensure that the codes would emerge in our time
in the altered text. Such
explanations are at the best appropriate for theological disputes rather than
for discussions involving mathematical statistics. Anyone who wishes to believe in the codes is of course entitled to that or to any other
beliefs. This article is though not
about beliefs but about the alleged "scientific"
or, more specifically, statistical proofs of the codes' genuineness.

The arguments discussed in this article show that neither WRR nor anybody else
among their supporters and code proponents have so far succeeded in
scientifically proving that ELS in the Bible are anything but accidental
sequences of letters which naturally occur in any sufficiently long text. WRR's argumentation has been debunked from various
viewpoints. It was shown that the statistical procedures employed by WRR were in
several ways contrary to the established rules of mathematical statistics. It was also demonstrated that the "distance" between ELS had been
defined by WRR in an unnatural manner, depriving that quantity of a meaningless
interpretation. It was indicated that the aggregate criteria of the ELS
proximity behave in a haphazard way, thus destroying the credibility of their
statistical conclusions from the standpoint of general scientific approach. It
was demonstrated that WRR had considerable "wiggle room," i.e. sufficient
freedom of arbitrary choices to fit their data to the desired outcome. It was indicated that the results of WRR's experiment, despite
considerable effort, could not be
reproduced by unbiased scientists. (The
only other experiment which allegedly confirmed the presence of a "code" in
the Bible was conducted by Harold Gans. Gans
is Director of Research at Aish Ha Torah, making
his living by promoting the "codes." Simon attempted to reproduce Gans's
experiment. Simon slightly changed
the procedure, aiming at the elimination of a possible ambiguity caused by the
"wiggle room." Namely, Simon included into consideration the names of all cities where the
"famous rabbis" were born, died, worked or studied, without any changes in
spelling or any addition of prefixes. Thus the "wiggle room" was
substantially reduced. The result? No trace of any code found in the Book of
Genesis. Conclusion? Gans'
results are most likely due to the "wiggle room" and therefore they cannot
be considered a confirmation of WRR's results. MBBK have also tried to
reproduce Gans' experiment, using a slightly different version of the data
lists, and found no code in Genesis either. (The details have been described in
MBBK's paper [8].)

As to WRR's epigones of all persuasions, such as Drosnin, Satinover, Jeffrey, etc.,
their work simply does not meet even the minimal scientific requirements to be
considered seriously.

It is therefore not surprising that in 1997 WRR and Drosnin were awarded the
"Ig Noble Prize" at a ceremony at Harvard (see The Ig Nobel Home Page) The Ig Noble Prize is awarded annually by a committee which includes real
Nobel price winners. It is given for "the discoveries which cannot and must
not be reproduced."

The publication of MBBK's
paper [8] deprived the code proponents of one of their most frequently used
argument, which suggested that the publication of their paper in 1994 in Statistical Science meant the approval of their claims by the
scientific community.

However, rather than admitting, at least
silently, the faults of their study, WRR organized instead the so-called
International Torah Code Society. They
conduct annual meetings of that society whose status in the scientific community
is not much different from that of the Flat Earth Society. To give a talk at
these meetings, the potential presenters must claim adherence to the Halakha,
i.e. to the tenets of the Orthodox Judaism.

On the
other hand, the Internet is full of websites where endless examples of alleged
codes in the Bible are demonstrated, mostly from a Christian standpoint, many of
them blatantly absurd, often revealing their authors' ignorance both of
mathematics and of Hebrew. Of
course, Rips, Witztum, and the International Torah Code Society as a whole
disdainfully dismiss all these publications as nonsense, while stubbornly
adhering to their own "discoveries." This gives the entire story a distinctively comic hue.

It seems reasonable to conclude that the
errors by WRR's and by their followers invalidate
their claims in regard to the alleged codes in the Bible and make their work an
epitome of what initially was bad science but eventually became pseudo-science.